Saint-Venant's principle

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15 Νοε 2013 (πριν από 3 χρόνια και 11 μήνες)

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1

SAINT VENANT’S PRINCIPLE



By

VIJAYA LAKSHMI EMANI.

MEEN


5330
.

2







INTRODUCTION
[3]


Saint
-
Venant’s principle is used to justify approximate solutions to
boundary value problems in linear elasticity.




For
example
, when solving problems involving bending or axial
deformation of slender beams and rods, one does not prescribe loads in
any detail. Instead, the resultant forces acting on the ends of a rod is
specified, or the magnitudes of point forces acting on a beam. Saint
Venant’s principle used to justify this approach.

3

DEFINITIONS
[2]

SAINT VENANT’S PRINCIPLE:

If a system of forces acting on a small portion of the surface of an
elastic body is replaced by another statically equivalent system of
forces acting on the same portion of the surface, the redistribution of
loading produces substantial changes in the stresses only in the
immediate neighborhood of the loading, and the stresses are essentially
the same in the part of the body which are at large distances in
comparision with the linear dimension of the surface on which the
forces are changed.


STATICALLY EQUIVALENT:

It means that the two distributions of forces have the same resultant
force and moment.

4

SAINT VENANT’S PRINCIPLE


MATHEMATICAL
FORM
[1]

Let S' and S'' be two non intersecting
sections both outside a sphere B. If
the section S'' lies at a greater
distance than the section S' from the
sphere B in which a system of self
equilibrating forces P acts on the
body, then:


U
R
' < U
R
"
, where U
R
' and
U
R
" are strain energies.

5

COMMON ENGINEERING INTERPRETATION OF THE
SAINT VENANT’S PRINCIPLE
[5]

“As long as the different approximations are statically equivalent , the resulting
solutions will be valid provided we focus on regions sufficiently far away from
the support.That is, the solutions may significantly differ only within the
immediate vicinity of the support.”





OR

“It is to say that the manner in which the forces are distributed over a region is
important only in the vicinity of the region.”



6





EXAMPLE
.
[6]

Surface loaded half
-
space.














Closed form expressions are known for the fields induced by many axi
-
symmetric
traction distributions acting on the surface of a half
-
space. For example, the fields down
the symmetry axis due to a uniform normal pressure acting on a circular region of
radius
a

are:



7

EXAMPLE CONTINUED


Where
P

is the resultant force acting on the loaded region. Similarly, the stresses due to
a Hertz pressure




Are,





Expand these in
a/z

and one will find that to leading order in
a/z

both
expressions are identical. Indeed, the leading order term is the stress induced by a point
force acting at the origin. Far from the loaded region, the two traction distributions
induce the same stresses, because the resultant force acting on the loaded region is
identical.


8


GENERAL SAINT VENANT BEAM PROBLEM


It is very complicated to solve the beam problem by considering some
definite distribution of surface forces on the end sections of the beam.
Hence we assume the validity of Saint
-
Venant’s principle.




So now the exact distribution of the surface forces on each end section
is replaced by another one that is statically equivalent
-
that is, one with
the same resultant R and the same moment M with respect to some
point O.

9

GENERAL CASES TO SOLVE USING SAINT VENANT
PRINCIPLE
[4]


There are four general classes of Saint
-
Venant problem corresponding to
various choices of end loading.

1. Extension if M =0 and n x P = 0

2. Torsion if P = 0 and n x M = 0

3. Bending by a torque if P = 0 and n . M = 0

4. Bending by a force if M = 0 and n . P= 0






10


BENDING OF A BEAM BY A TORQUE


Consider a cylindrical beam of
arbitrary cross section and a length
2L.On the end sections are applied
normal forces, distributed in such a
way that on each end section the
resultant is zero, and the total
moment is tangent to the section.


11

SYSTEM OF EQUATIONS TO BE CONSIDERED FOR
DEFORMATION CALCULATION
[4]


We assume that the torque applied at the end section z = L is M
j
. Then
for equilibrium we must have a torque

M
j

on end section z =
-
L. The
system of Equilibrium equations to be satisfied are then


1.


. T = 0
}in the volume V of the beam.

2.


x S x


= 0


3. n . T =0 on the lateral surface, where n.
k
=0


4.
k

. T x
k

= 0

5.
∫s dS

k

. T = 0 on the surfaces z=
±
L r being equal to xi+yj.

6.
∫s dS r x (
k

. T) =
±

M
j

12

DEFORMATION CALCULATION


From the equations n . T =0 and
k

. T x
k

= 0


The stress dyadic has the simple form as follows:


T = T
33

kk
.




Also from


. T = 0


we get

/

z( T
33
) = 0.This shows that T
33

depends on x and y only.



Thus the Strain dyadic is given by :


S =
1

[
kk

-


(
ii

+
jj
)] T
33
(x,y).


E




13


DEFORMATION CALCULATION


S

=

+ ii



The compatibility equation


x S x


gives the following four
equations:


=0






Which implies that T
33

= Ax + By+ C


where A,B and C are arbitrary constants to be determined from the
boundary conditions. From condition 5.∫s dS
k

. T = 0 we get


∫s dS ( Ax + By + C) = 0.


Also from ∫s ds r =0 we have

= 0

and

y = 0

.


From ∫s dS r x (
k

. T) =
±

M
j
























y
x
kk
E
2
2
2
2
1

)
(
2
2
2
2
ji
ij
jj
x
y















y
x
2
T
33



33
2
2
T
x


0
33
2
2



T
y
0
33
2




T
y
x
0
33
2
2
2
2














T
y
x

dSx
s

dS
14

DEFORMATION CALCULATION

∫s dS r x (
k

. T) =
±

M
j

we obtain on z=L,

-
j

The exact solution for any cross section is T=
-
M/I(x
kk
)

Where I =

s
dS x
2

The strain dyadic is

Or


=
-


From this we get the displacement vector is





Mj
By
Axy
dS
i
Bxy
Ax
dS
s
s






)
(
)
(
2
2




1
1






kk
x
EI
M
S




r
x
zk
x
EI
M
S








1






















2
1
1
2
2
r
z
i
xr
xzk
EI





























2
2
1
2
2
ir
xr
kxz
iz
EI
M
s
















kxz
xy
j
y
x
z
i
EI
M


2
2
2
2
1
15

PRACTICAL APPLICATIONS AND LIMITATIONS


APPLICATIONS

Saint
-
Venant’s principle is used to
justify approximate solutions to
boundary value problems in linear
elasticity.




The principle of Saint Venant
allows us to simplify the solution
of many problems by altering the
boundary conditions while keeping
the systems of applied forces
statically equivalent. A satisfactory
approximate solution can be
obtained.



LIMITATIONS

The Saint Venant’s Principle, as
enunciated in terms of the strain
energy functional, does not yield
any detailed information about
individual stress components at any
specific point in an elastic body.
But such information is clearly
desired .

Saint
-
Venant himself limited his
principle to the problem of
extension, torsion and flexure of
prismatic and cylindrical bodies.





16

HOMEWORK

1. Write a one page essay on Saint Venant’s Principle.


2. A cylindrical beam of length 2L has an arbitrary cross section. The
beam is subjected to a force R
k

acting at a point P of the boundary of
the end section z=L, and to a force

R
k

acting at a point p
'
symmetrical to P with respect to plane z=0. solve this problem using
Saint Venant’s principle and determine the deformation.

17



REFERENCES


1.

Foundations of Solids Mechanics” by Y.C.Fung.

2.“Elasticity Theory and Applications” Second Edition by
Adel S. Saada.

3.

Theory of Elasticity” by Timoshenko and Goodier.

4.

Introduction to Elasticity” by Gerard Nadeau.

5.

Theory of Elasticity” by Southwell.

6.http://www.engin.brown.edu/courses/En222/Notes/elastpri
ns/elastprins.htm

7.http://www.engin.brown.edu/courses/En224/svtorsion/svto
rsion.html